Wednesday, April 16, 2014

Basics of Geometry in Sanskrit || संस्कृतम ||


|| संस्कृतम ||




|| वदतु संस्कृतम || पठतु संस्कृतम ||



|| संस्कृत अध्ययन कार्ये स्वागतम् अस्तु ||

Welcome to learning learning Sanskrit



|| Sanskrit_Adhyayan || Sanskrit_Sanskruti || Bharatiya_Knowledge_Traditions || Sanskrit_Jagruti ||




१) अस्तित्वं भासते यस्य न शक्यं मापनं खलु |

निरकारोsपि साकारो स बिन्दुरिति कथ्यते ||



Meaning : A place whose existence is experienced/seen but which can not be measured is called as a Point.



Explanation : A point is a location in space.A point has no length,width or height, it just specifies an exact location.







२) बिन्दुनाम य: समूह: स्यात घनविस्तार: वर्जित: |

दीर्घाकार: स भूमित्याम रेखाखंड इति स्मृत: ||



Meaning : A line is a straight one-dimensional geometric figure formed by collection of points having no thickness and volume (and extending infinitely

in both directions ).





३) भवतो दैर्घ्यविस्तारौ घनता नैव विद्यते |

प्रुष्टभागसमम् रूपं भूमित्याम् प्रतलं हि तत् ||



Meaning : An (imaginary) flat surface that is infinitely large and with zero thickness or volume is defined as 'plane' in geometry.







४) रेखाया: प्रतलस्यापि द्वयम् यदि परस्परम् |

न संस्प्रुश्यति भूमित्याम् प्रतलम् हि तत् ||



Meaning : Two lines (in a plane) or two planes that do not intersect or meet are called parallel (lines and planes respectively).







५) प्रमाणम् नवतिर्यस्य काटकोण: स: उच्यते |

ततोSधिको विशाल: स्यात् तन्यूनो लघुरुच्यते ||



Meaning : The traingle that has (one of its interior )angle measuring 90 degrees is called as 'right traingle'.The one greater then that (that has one angle that measures more than 90 degrees)is called as 'obtuse triangle' and the one lesser than that( that has all interior angles measuring less than 90 degrees) is called as 'acute traingle'.







६) तिस्रो भुजा समा यस्य समानभुज उच्यते |

तथा द्वयम् समानम् स्यात् स समद्विभूज भवेत् ||



Meaning : The triangle whose three sides are equal is called as 'equilatera triangle'. Similarly, the triangle whose two sides are equal is called as 'isosceles triangle'







७) अंशयोगत्रिकोणस्य शतं चाशीति लक्षणम् |

चतुर्भुजस्य कोणानां योगो द्विगुणितो भवेत् ||



Meaning : The sum of the angles of a triangle is 180 degrees ,while the sum of the angles of a square is twice the 180 degrees i.e. 360 degrees.



८) त्रिभुजस्य फलशरीरम् समदलकोटीभुजार्धसंवर्ग: |

समपरिणाहस्यार्धम् विष्कम्भार्धहतमेव वृत्तफलम् ||



Meaning : Area of a triangle is ((1/2)*(perpendicular height)*base),while area of a cirlcle is (pi*r*r).



Explaination : फलशरीरम् - area, समदलकोटी - perpendicular,

समपरिणाह - circumference, विष्कम्भ - diameter

Area of a circle = (circumference/2)* (diameter/2)

=((2*pi*r)/2) * (2r/2)

=2*pi*r





९)चतुरधिकं शतं अष्टगुणं द्वासष्टीस्तथा सहस्त्रानाम |

अयुतद्वयं विष्कम्भ: स्या सन्नो वृत्तपरिणाह: ||



Meaning : The figure whose diameter is 20,000 has a circumference of 62,832. Explaination : Aryabhatt has given value of mathematical symbol π ('pi') in this shloka.

चतुरधिकं शतं अष्टगुणं - 104 * 8,(=832)

द्वासष्टीसहस्त्रानाम = 62,000, अयुतद्वयं = 20,000

विष्कम्भ: - diameter , वृत्तपरिणाह: - circuference

So, π = 62,832/20,000 = 3.1416



१०) वर्ग: समचतुरस्त्र फलम् च सदृशद्वयस्य संवर्ग:| सदृशत्रयसम्वर्गो घनस्तथा द्वादशाश्रि: स्यात् ||

Meaning : The figure with 4 equal sides is called as varg(square).Similarly,the multiplication of two equal numbers is callled as (its) varg(square).
The multiplication of three equal numbers is callled as (its) ghan(cube).Similarly,the figure with 12 equal sides is called as a ghan (cube).



उदाहरनानि (Examples) :


११)क्षेत्रस्य समकोणस्य दैर्घ्यम् यदि चतुर्दश | दैघ्यार्ध-मात्राविस्तार: फलम् परिमितिम् वद ||


Meaning : Tell (me) the area and circumference of a farm with length equal to 14 units and width equal to half of the length.



१२)कूपस्य सप्त विष्कभो निम्नता पंचविन्शति: | परिपूर्णे हि कूपेSस्मिन कियद्वारी भविष्यति? ||

Meaning : The diameter of a well is 7 units and its height is 25 units.What amount of water can completely fill the well?





संख्यारचना (Positional Number System)





१) एकं द्वे त्रिणि चत्वारि पञ्च षट सप्त चाष्ट च |

नव शून्यं दशांका स्यु: संख्या-लेखन-हेतवे ||



Meaning : 1,2,3,4,5,6,7,8,9,0 are the symbols to write numbers.



२) तस्मात् संयुज्य तान् अंकान् स्थानमानानुसारत: |

वामतो गति: अंकानाम् ज्ञात्वा संख्या च लिख्यते ||



Meaning : Hence,by combining these number-symbols (and) by knowing that their (local/positional) value increases (from right) towards left,

a number is written.